That is part of the clutch pack retaining clip. You're probably a peg leg now. You can try doing a quick launch on loose ground and see if both wheels spin.
That's not all of the retaining clip, so someone has pulled out the actual end retainer previously. You may want check and see if you have a clip on either side.
C clip and clutch pack retaining clip are two entirely different things.
How many diff fluid changes have you done on this truck?
It is ok to drive with it, just realize that you probably don't have anti-spin/LSD, and, there may be more parts coming.
So how does all of this fit with the play you're experiencing at the pinion yoke? They may be unrelated, as the clutch pack retaining clip is a very common problem. Or it could be that an extreme event caused both problems. You'll never know and really doesn't matter.
I'd recommend you go ahead with the u joint replacement. Check for pinion yoke play with the drive shaft off.
Also take a look at the backing plates on both rear wheels and see if you have gear oil leaking. You mentioned adding gear oil and you could have a leaking seal.

 

I just had my rear differential completely rebuilt cuz of a shudder taking off at red light or going into reverse. Mechanic didnt feel it on short test drive, so I said... Humor Me... pull the cover. Annnd.....

 

That looks awful familiar......

 

Welcome to the club. Source an open diff from a compatible model year truck and have a gear shop build it up. I say open diff because it's immune to that failure, so they're usually a good bet on having a good pumpkin.

If left untreated those clips can chew far enough into the carrier bearing cap bolt holes to cause the carrier to break free.

Here's pics of mine when it blew up.